sometimes there is a need to feed Mathematica with extra knowledge
so for your case
Unprotect[InverseFunction];
InverseFunction[Conjugate, 1, 1][x_] := Conjugate[x];
Protect[InverseFunction];
And now
Solve[Conjugate[x]==I,x]
and you will get
{{x -> -\[ImaginaryI]}}
yehuda